Problem: Divide the polynomials. The form of your answer should either be $p(x)$ or $p(x)+\dfrac{k}{x-2}$ where $p(x)$ is a polynomial and $k$ is an integer. $\dfrac{x^3+6x^2-5x}{x-2}=$
Solution: Usually, there are many different ways to divide polynomials. Here, we will use the method of polynomial long division. Notice that the expression in the numerator is missing a constant (degree $0$ ) term. To avoid any confusion, let's add that term as $0$. $\begin{array}{r} x^2+\phantom{1}8x+11 \\ x-2|\overline{x^3+6x^2-\phantom{1}5x+\phantom{2}0} \\ \mathllap{-(}\underline{x^3-2x^2\phantom{+15x+20}\rlap )} \\ 8x^2-\phantom{1}5x+\phantom{2}0 \\ \mathllap{-(}\underline{8x^2-16x\phantom{+20}\rlap )} \\ 11x+\phantom{2}0 \\ \mathllap{-(}\underline{11x-22\rlap )} \\ 22 \end{array}$ We found that the quotient is $x^2+8x+11$ and the remainder is $22$ : $\dfrac{x^3+6x^2-5x}{x-2}=x^2+8x+11+\dfrac{22}{x-2}$